Download x - Miss Hudsons Maths

Angle
Geometry
Measuring Angles
 AOB = 27o
AOC = 122°
C
B
A
O
Naming Angles
Angles are named using the three letters that form
them.
The middle of the three letters is the vertex of the
angle (pointy bit)
D
DEF or DÊF or FED or FÊ D
E
F
Types of angle
Acute angle
(less than
90°)
Right Angle
(90°)
Obtuse angle
(between 90°
and 180°)
Straight angle
(180°)
Reflex angle
(more than 180°)
Complementary angles add up to 90°
So the complement of 40° 50°
is
Supplementary angles add up to 180°
The supplement of 40° is140°
CO
1S0
130°
x
x = 50°
Adjacent angles on a straight
line sum to 180°
Vertically opposite angles are equal
x
70°
x
80° 120°
110°
x = 70°
Angles at a point sum to
360°
x = 50°
transversal
 
 110

 x °
50° 
 x
 110°
 x
x = 110°
Corresponding angles on
parallel lines are equal
x = 50°
Alternate angles on parallel
lines are equal
x = 70°
Co-interior angles on parallel
lines sum to 180°
x
50°
x = 60°
70°
Angles in a triangle sum to 180°
demo
x=
50°
Base angles of an isosceles
triangle are equal
50°
x
x = 60°
Each angle in an equilateral
triangle is 60°
x
50°
x
70°
x = 60°
Angles in a quadrilateral sum
to 360°
100°
x
120°
x = 120°
Exterior angle of a triangle equals the
sum of the 2 interior opposite angles
80°
»
Parallelogram
I
□
□ I □
Rectangle
Rhombus
»
I
I
□
□
II
»
□ I
II
Types of Quadrilaterals
□ I
□
Square
»
Trapezium
Kite
Angles on Parallel Lines
transversal






110°
x
50°




x
110°
x
x = 110°
Corresponding angles on
parallel lines are equal
x = 50°
Alternate angles on parallel
lines are equal
x = 70°
Co-interior angles on parallel
lines sum to 180°
Angles on parallel lines
Types of Triangles
(according to sides)
Scalene - no equal sides
Isosceles - 2 equal sides
Equilateral - 3 equal sides
Types of Triangles (according to angles)
Acute angled triangle - all angles less than 90°
Right angled triangle - one 90° angle
Obtuse angled triangle - one angle  90°
Polygons
Can you remember the special
names for polygons that have
five, six, eight and ten sides
A polygon is a closed figure
made up of straight sides.
Names of polygons:
Number of sides
Name
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
8
Octagon
10
Decagon
A regular polygon has all it’s sides equal and all it’s
angles equal. E.g. a square is a regular quadrilateral.
Interior and Exterior angles
The blue angles are interior
angles of the triangle.
The grey angles are exterior
angles of the triangle.
Exterior Angles of Polygons
The sum of the exterior angles of any polygon is always 360°
A regular polygon is one with all sides equal in length and
all angles equal in size.
So in a regular polygon with n sides, each exterior angle is
360 °
n
The sum of the exterior angles = 360°
Each exterior angle of this regular
hexagon is 360 ÷ 6,
so b = 60°
Interior Angles of Polygons
The sum of the interior angles for a polygon with n sides is
(n – 2) x 180o
example: Find x
Number of sides: n = 5
Sum of interior angles
= (5 – 2) x 180
= 3 x 180
= 540°
x + 120 + 160 + 45 + 170 = 540
x + 495
= 540
x = 45°
To find the size of each interior angle of a regular polygon,
divide the sum by the number of sides.
eg: Find the size of each interior angle of a regular octagon
soln: n = 8
Cool,
Sum of interior angles = (8 - 2) x 180°
eh?
= 6 x 180°
= 1080°
Each interior angle = 1080 ÷ 8
= 135°
Constructions
1)
To Bisect an Angle
A
P
.
. .
Method:
• With centre B, scribe an arc to cut the two arms.
Label these points P & Q
• From P and Q draw arcs to intersect at a point
which we will label M
• Join M to B
M
B
QC
2)
To Bisect a Line Segment
P
.
.
A
Q
B
Method:
• From A and B scribe arcs to intersect at
2 points, P and Q
• Join P to Q
3)
A
To construct an angle of 90° at a given point P on a line
segment AB
Method:
Q
• With centre P, scribe an arc to cut the
line segment at M and N
• From M and N scribe 2 arcs to meet
M
N
B
at Q
•
P
• Join Q to P
4) To construct an angle of 90° from a given point P off a
line
Method:
segment
• P AB
• With centre P, scribe an arc to cut the
line segment at M and N
M
N B
A
• From M and N scribe arcs to meet at Q
Q
• Join P to Q
5)
To Construct a Triangle
Given the lengths of the 3 sides, say AB = 5 cm,
AC = 4 cm and BC = 3 cm
Method:
• Draw one of the sides, measuring it with a ruler, say AB
C
• With centre A scribe an arc of length 4 cm
• With centre B scribe an arc of length 3 cm
• Where the 2 arcs meet is point C
• Join C to A
A
• Join C to B
5 cm
B
6)
7)
X
To construct a Hexagon
Method:
• Draw a circle
• With the same radius, step the radius
.
off, around the circumference
• Join the points with straight line
segments to form a regular hexagon
To Construct an Angle of 60° on a given line segment XY
Method:
W
• With centre X and any radius, scribe an
arc which cuts XY at Z
Z
Y • With centre Z and using the same radius,
scribe an arc to cut the first one at W
• Join W to X. Angle WXY is 60°
 Parts
of the Circle
tangent
Sector
Compass Directions
N
NW
NNW NNE
WNW
NE
ENE
W
E
WSW
SW
ESE
SSW SSE
S
SE
N to E is 90°
N to NE is 45°
N to NNE is 22.5°
Bearings
A bearing is a direction.
It is always measured from North in a clockwise direction.
It always consists of 3 digits before the decimal point.
eg 135°, 067°, 036.5°
The compass directions as bearings are:
000
N
°
NW NNW
315°
NE
022.5 045°
337.5 NNE
°
°
067.5°
WNW
ENE
292.5°
W
E090
270
WSW
° 247.5
ESE
112.5°°
°
SW
SSW 157.5
SSE 135
225 202.5
SE
°
°°
°
180
S
°
Bearings Example N
252°
Find the bearing of the boat from the lighthouse.
So the bearing is 252°